- Paola Bonacini

“On the lifting problem in P^4 in characteristic p”

Abstract: here.

- Jarosław Buczyński

“Special lines on contact Fano manifolds”

Abstract: I will explain what is a contact Fano manifold X and why we are interested in studying it. The main conjecture in the subject (due to LeBrun and Salamon) is that X is always a homogeneous space, one of the adjoint varieties. The main tool from algebraic geometry to study the conjecture is the theory of minimal rational curves, i.e., in this case, the rational curves on X with minimal degree measured by the intersection with the anti-canonical divisor (they are called lines, or contact lines). A general line is smooth and standard, i.e,. it has a particularly simple splitting of the tangent bundle to X restricted to the curve. Working on contact Fano manifolds would go very smoothly, if all lines were smooth and standard. In our work we bound the dimensions of spaces of singular and non-standard lines. Specifically, we prove they are both at least codimension 2 in the space of all lines. The statement about non-standard lines was earlier claimed by Kebekus and used by him to prove the irreducibility of the space of lines through a fixed general point. However, his proof contained a gap.

- Luca Chiantini

“The level of stability of ACM bundles of rank 2 on surfaces”

Abstract: For a general surface in P^3 of degree d>1, the existence of ACM bundles which are (indecomposable and) stable is well known. We study in particular the existence of such bundles with high level of stability (i.e. whose initialized twist has a big first Chern class). We show that ACM rank 2 bundles with high level of stability have first sections which vanish in sets of points, whose position is far from being uniform.

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- Maria Lucia Fania

“On the Hilbert scheme of varieties defined by maximal minors”

Abstract: In this talk I will report on a joint with Daniele Faenzi. We compute the dimension of the Hilbert scheme of subvarieties of positive dimension in projective space which are cut by maximal minors of a matrix with polynomial entries.

- Elisa Gorla

“Universal Groebner bases for ideals of maximal minors”

Abstract: Bernstein, Sturmfels and Zelevinsky proved in 1993 that the maximal minors of a matrix of variables form a universal Groebner basis. We present a very short proof of this result, along with a broad generalization to matrices with multi homogeneous structures. Our main tool is a rigidity statement for radical Borel fixed ideals in multigraded polynomial rings. This is joint work with A. Conca ed E. De Negri (University of Genoa).

- Laurent Gruson

“Non-existence of globally generated rank-two locally free modules of Chern classes (5,12) over the projective space”

Abstract: Let P be the projective space. In the last Vector Bundles Day meeting, Ellia gave a list of globally generated, rank 2, locally free O_P -modules of low Chern classes, but he left open the existence of the highest numerical example, that of the title. The answer to this question is no. The usual Beilinson spectral sequence produces in this case a double structure on a Cohen-Macaulay curve of degree 6 and genus 3, and some trouble is caused by the possible non-lci (triple) points of this curve.

- Johannes Huisman

“Chern-Stiefel-Whitney classes of real vector bundles”

Abstract: Let X be a real algebraic variety and F a real vector bundle over X. I will define Chern-Stiefel-Whitney classes of F with values in certain hypercohomology groups on the quotient topological space X(C)/G, where G is the Galois group of C/R. These classes unify the ordinary characteristic classes in the sense that they induce the Chern classes of F(C), on the one hand, and the Stiefel-Whitney classes of F(R), on the other hand. The construction sheds a seemingly new light on the fact that the mod-2 cohomology ring of a real Grassmannian is the reduction modulo 2 of the integral cohomology ring of a complex Grassmannian after dividing all degrees by 2.

- Monica Idà

“Syzygies, fat points and the tangent bundle”

Abstract: If C is a rational curve in the projective plane, it is natural to look at the pull back of the tangent bundle of P^2 to the normalization of C, and to try to relate its splitting to the degree and singularities of the curve; in other words, to relate the syzygies of a parameterization of C to the singular points of C. I'm going to talk about joint work with A.Gimigliano and B.Harbourne on this problem and also on what first motivated us, that is, minimal free resolution of plane fat point schemes; the regularity of such a scheme can in fact be perturbed by a "too high secant" rational curve.

- Tomas Kozir

“Determinantal Representations of Cubic Curves”

Abstract: For every (irreducible) cubic curve all nonequivalent determinantal representations are explicitly constructed. We can tell which and how many of these representations are self-adjoint. We give a new proof of the criteria when a self-adjoint determinantal representation is definite; or equivalently, when it is a LMI representation of a spectrahedron on which semidefinite programming can be performed. Joint with Anita Buckley.

- Simone Marchesi

“Buchsbaum vs Instantons bundles”

Abstract: The goal of this talk is to give a (negative) answer to the following question: given a rank 2 bundle over P^3, is such bundle k-Buchsbaum if and only if it is a instanton bundle of quantum number k? In the first part of the talk we will recall the definitions and properties we need and in the second part we will prove that each k-instanton is actually k-Buchsbaum but the contrary is false, giving a classification of 3-Buchsbaum bundles (the cases 1 and 2 were already known). This is a joint work with Marcos Jardim and Juan Pons.

- Giorgio Ottaviani

“Euclidean normal bundle and ED degree of an embedded algebraic variety”

Abstract: The Euclidean Distance (ED) degree of an affine variety X

embedded in a euclidean space is the number of critical points of the euclidean distance of X from a general point.

The ED degree is well defined for projective varieties as well, by looking at their affine cones, in this case it can be computed as the top Segre class of the euclidean normal bundle of X, following Catanese and Trifogli, under some transversality assumptions. We show the relation of ED degree with polar classes and its nice behaviour under projective duality. We discuss the relation with rank r best approximation of tensors, where the transversality assumptions are not verified, and some connected open problems. This talk arises from three joint papers with Draisma, Friedland, Horobet, Spaenlehauer, Sturmfels, Thomas.

- Kristian Ranestad

“Varieties of sums of powers of a cubic 4-folds”

Abstract: The variety VSP(f,10) of presentations of a general homogeneous cubic form f in 6 variables as a sum of 10 powers of linear forms has the structure of a hyperkahler 4-fold. I shall discuss this result and how Noether Lefschetz divisors in the moduli space of Hyperkahler 4-folds correspond to divisors of cubic fourfolds that are not Noether Lefschetz. I report on work with Iliev and Voisin.

- Lidia Stoppino

“Stability and geometry of relative hypersurfaces”

Abstract: Let P be a projective bundle over a smooth curve B. The nef and pseudoeffective cones of P have beeen classically computed by Miyaoka and Nakayama. In this talk we will present an instability result on the divisors whose class lies in an intermediate cone contained in PEff(P)\Nef(P). This result implies that these divisors are extremely singular: in particular we will show a strong upper bound on their log canonical treshold. We will eventually discuss some cases when explicit computations can be made. This is a joint work with Miguel Angel Barja.

- Fabio Tanturri

“Degeneracy loci of sections of the twisted cotangent bundle”

Abstract: Given a morphism F between vector bundles on P^N=P(V), its degeneracy locus X is made up of points in which the morphism has not maximal rank. In order to parameterize such degeneracy loci, one can look at the union of the irreducible components of the Hilbert Scheme H_X containing the general X's. I will focus on the case in which the morphism is given by m global sections of Omega(2), the twisted cotangent sheaf on P(V). In this case, making use of the Kempf-Weyman's method for computing syzygies via resolutions of singularities, H_X can be proved to be birational to the Grassmannian Gr(m,Lambda^2(V)), when 3 < m < N+1. In the cases m=3 and m=2 it is possible to give a precise geometric characterization of a general X and explain why the previous birationality does not occur.

- Luca Ugaglia

“On cubic elliptic varieties”
Abstract: here.